The Lorentz Group: Circles and hyperbolas

Posted by doug
The Lorentz group has one subgroup of 3D rotations, and another for inertial reference frame boosts where time effectively rotates into a spatial dimension.
The spacial rotations are in yellow, while the boosts are in red. The hyperbolas can cover all of spactime while the rotations are limited to a circle about the origin. Out at infinity, there will be two points that approach the yellow circle. They split so that one set of red points can be there are the creation and annihilation of the yellow points, and another set can greet the yellow points when they are furthest apart, about to change directions.
q_graph -box 3 -dir trig -out circle -command 'g_u1_n 4 1 2 3 1000' -color yello -command 'g_u1h_n 4 1 2 3 4000' -color red [note: the function for generating the hyperbola has not been released, and its name will probably change]

for the circle: \frac{q}{\sqrt{q q*}} = \frac{(t, \vec{R})}{t^2 + R^2}
for the hyperbola: \frac{q}{\sqrt{\pm scalar(q q)}} = \frac{(t, \vec{R})}{t^2 - R^2}